Difference between revisions of "Saturated models"
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In a recent paper Nurkhaidarov and Schmerl 2011 proved the following: Let κ be regular, uncountable, and such that κ<κ=κ. For each completion T⊇PA let MκT be the saturated model of T of cardinality κ. There is a set T of completions of PA, such that |T|=2ℵ0 and for all T,T′∈T, if T≠T′, then Aut(MκT)≇Aut(MκT′). The following question is left open: Are there T and T′ such that T≠T′ and Aut(MκT)≅Aut(MκT′)? | In a recent paper Nurkhaidarov and Schmerl 2011 proved the following: Let κ be regular, uncountable, and such that κ<κ=κ. For each completion T⊇PA let MκT be the saturated model of T of cardinality κ. There is a set T of completions of PA, such that |T|=2ℵ0 and for all T,T′∈T, if T≠T′, then Aut(MκT)≇Aut(MκT′). The following question is left open: Are there T and T′ such that T≠T′ and Aut(MκT)≅Aut(MκT′)? | ||
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+ | == Elementary cuts == | ||
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+ | A cut in a saturated model is balanced if its upward and downward cofinalities are equal to the cardinality of the model. Since there are continuum theories of pairs (N,M), where N⊨PA and M⊨PA, and N is an elementary end extension of M, it follows that for every saturated model N there are continuum non elementarily equivalent, hence nonisomorphic, pairs (N,M), where M is an elementary balanced cut of N. For a saturated N, what is the maximum number of such pairs? | ||
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+ | For an almost complete classification of elementary cuts in saturated models of PA see James Schmerl ''Elementary cuts in saturated models of Peano arithmetic''. Notre Dame J. Form. Log. 53 (2012), no. 1, 1–13, |
Revision as of 09:24, 18 January 2013
The automorphism group
In a recent paper Nurkhaidarov and Schmerl 2011 proved the following: Let κ be regular, uncountable, and such that κ<κ=κ. For each completion T⊇PA let MκT be the saturated model of T of cardinality κ. There is a set T of completions of PA, such that |T|=2ℵ0 and for all T,T′∈T, if T≠T′, then Aut(MκT)≇Aut(MκT′). The following question is left open: Are there T and T′ such that T≠T′ and Aut(MκT)≅Aut(MκT′)?
Elementary cuts
A cut in a saturated model is balanced if its upward and downward cofinalities are equal to the cardinality of the model. Since there are continuum theories of pairs (N,M), where N⊨PA and M⊨PA, and N is an elementary end extension of M, it follows that for every saturated model N there are continuum non elementarily equivalent, hence nonisomorphic, pairs (N,M), where M is an elementary balanced cut of N. For a saturated N, what is the maximum number of such pairs?
For an almost complete classification of elementary cuts in saturated models of PA see James Schmerl Elementary cuts in saturated models of Peano arithmetic. Notre Dame J. Form. Log. 53 (2012), no. 1, 1–13,